3.1065 \(\int \frac{1}{x \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=197 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{8 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]

[Out]

-ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)]/(4*2^(3/4)) - ArcTan[1 + (4 - 6*x^2)^(1/4)]/(
8*2^(1/4)) + ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)]/(8*2^(1/4)) - ArcTanh[(2 - 3*
x^2)^(1/4)/2^(1/4)]/(4*2^(3/4)) + Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt
[2 - 3*x^2]]/(16*2^(1/4)) - Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3
*x^2]]/(16*2^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.447456, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{8 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

-ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)]/(4*2^(3/4)) - ArcTan[1 + (4 - 6*x^2)^(1/4)]/(
8*2^(1/4)) + ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)]/(8*2^(1/4)) - ArcTanh[(2 - 3*
x^2)^(1/4)/2^(1/4)]/(4*2^(3/4)) + Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt
[2 - 3*x^2]]/(16*2^(1/4)) - Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3
*x^2]]/(16*2^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.2836, size = 178, normalized size = 0.9 \[ \frac{2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{32} - \frac{2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{32} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{8} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{16} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{16} - \frac{\sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)

[Out]

2**(3/4)*log(-2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/32 -
2**(3/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/32 - 2
**(1/4)*atan(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/8 - 2**(3/4)*atan(2**(1/4)*(-3*x**
2 + 2)**(1/4) - 1)/16 - 2**(3/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) + 1)/16 - 2*
*(1/4)*atanh(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/8

_______________________________________________________________________________________

Mathematica [C]  time = 0.281857, size = 139, normalized size = 0.71 \[ \frac{66 x^2 F_1\left (\frac{7}{4};\frac{3}{4},1;\frac{11}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )}{7 \left (2-3 x^2\right )^{3/4} \left (3 x^2-4\right ) \left (33 x^2 F_1\left (\frac{7}{4};\frac{3}{4},1;\frac{11}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+16 F_1\left (\frac{11}{4};\frac{3}{4},2;\frac{15}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+6 F_1\left (\frac{11}{4};\frac{7}{4},1;\frac{15}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(66*x^2*AppellF1[7/4, 3/4, 1, 11/4, 2/(3*x^2), 4/(3*x^2)])/(7*(2 - 3*x^2)^(3/4)*
(-4 + 3*x^2)*(33*x^2*AppellF1[7/4, 3/4, 1, 11/4, 2/(3*x^2), 4/(3*x^2)] + 16*Appe
llF1[11/4, 3/4, 2, 15/4, 2/(3*x^2), 4/(3*x^2)] + 6*AppellF1[11/4, 7/4, 1, 15/4,
2/(3*x^2), 4/(3*x^2)]))

_______________________________________________________________________________________

Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ( -3\,{x}^{2}+4 \right ) } \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)

[Out]

int(1/x/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (3 \, x^{2} - 4\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x),x, algorithm="maxima")

[Out]

-integrate(1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.247947, size = 381, normalized size = 1.93 \[ \frac{1}{128} \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (4 \, \sqrt{2} \arctan \left (\frac{2}{8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 4}}\right ) - \sqrt{2} \log \left (8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2\right ) + \sqrt{2} \log \left (8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 2\right ) + 4 \, \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2 \, \sqrt{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2} + 2}\right ) + 4 \, \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8} - 2}\right ) - \log \left (4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right ) + \log \left (-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x),x, algorithm="fricas")

[Out]

1/128*8^(3/4)*sqrt(2)*(4*sqrt(2)*arctan(2/(8^(1/4)*(-3*x^2 + 2)^(1/4) + sqrt(2*s
qrt(2)*sqrt(-3*x^2 + 2) + 4))) - sqrt(2)*log(8^(1/4)*(-3*x^2 + 2)^(1/4) + 2) + s
qrt(2)*log(8^(1/4)*(-3*x^2 + 2)^(1/4) - 2) + 4*arctan(2/(8^(1/4)*sqrt(2)*(-3*x^2
+ 2)^(1/4) + 2*sqrt(8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + sqrt(2)*sqrt(-3*x^2 +
2) + 2) + 2)) + 4*arctan(2/(8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + sqrt(-4*8^(1/4)
*sqrt(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2)*sqrt(-3*x^2 + 2) + 8) - 2)) - log(4*8^(1
/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2)*sqrt(-3*x^2 + 2) + 8) + log(-4*8^(1/4
)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2)*sqrt(-3*x^2 + 2) + 8))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{3 x^{3} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}} - 4 x \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)

[Out]

-Integral(1/(3*x**3*(-3*x**2 + 2)**(3/4) - 4*x*(-3*x**2 + 2)**(3/4)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.257703, size = 284, normalized size = 1.44 \[ -\frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (-\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{32} \cdot 4^{\frac{1}{8}} \sqrt{2}{\rm ln}\left (4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) + \frac{1}{32} \cdot 4^{\frac{1}{8}} \sqrt{2}{\rm ln}\left (-4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) - \frac{1}{8} \cdot 4^{\frac{1}{8}} \arctan \left (\frac{1}{4} \cdot 4^{\frac{7}{8}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{1}{16} \cdot 4^{\frac{1}{8}}{\rm ln}\left ({\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) + \frac{1}{16} \cdot 4^{\frac{1}{8}}{\rm ln}\left (-{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x),x, algorithm="giac")

[Out]

-1/16*4^(1/8)*sqrt(2)*arctan(1/8*4^(7/8)*sqrt(2)*(4^(1/8)*sqrt(2) + 2*(-3*x^2 +
2)^(1/4))) - 1/16*4^(1/8)*sqrt(2)*arctan(-1/8*4^(7/8)*sqrt(2)*(4^(1/8)*sqrt(2) -
2*(-3*x^2 + 2)^(1/4))) - 1/32*4^(1/8)*sqrt(2)*ln(4^(1/8)*sqrt(2)*(-3*x^2 + 2)^(
1/4) + sqrt(-3*x^2 + 2) + 4^(1/4)) + 1/32*4^(1/8)*sqrt(2)*ln(-4^(1/8)*sqrt(2)*(-
3*x^2 + 2)^(1/4) + sqrt(-3*x^2 + 2) + 4^(1/4)) - 1/8*4^(1/8)*arctan(1/4*4^(7/8)*
(-3*x^2 + 2)^(1/4)) - 1/16*4^(1/8)*ln((-3*x^2 + 2)^(1/4) + 4^(1/8)) + 1/16*4^(1/
8)*ln(-(-3*x^2 + 2)^(1/4) + 4^(1/8))